where \(\alpha \) , \(\beta \) , \(\gamma \)  and \(\varphi \)  are the
spacetime-dependent metric perturbation variables. In WI the
evolution of field fluctuations \(\delta \phi \) , which is effectively
described by a stochastic evolution, is determined by a Langevin-like
equation [1]}, [2]}. In the case of
DBIWI, perturbing the covariant equation (REF ) around the
background, i.e., \(\Phi ({\bf x},t) = \phi (t) + \delta \phi ({\bf x},t)\) , the corresponding Langevin equation for \(\delta \phi \)  reads
\(\nonumber \delta \ddot{\phi }&+ 3 H \ \left(1 +Q-\epsilon _{s}\right)\delta \dot{\phi }+ H^{2} \left\lbrace  z^{2}+ \frac{3\dot{f}}{2Hf} \left[ (1-c_{s}^{2}) (1+Q) - \epsilon _{s}\right]-\frac{1-3c_{s}^{2}+2c_{s}^{3}}{H^{2}}\left(\frac{{f^{\prime }}^{2}}{f^{3}}- \frac{f^{\prime \prime }}{2f^{2}} \right) +\frac{c_{s}^{3}V^{\prime \prime }}{H^{2}} \right\rbrace  \delta \phi \\ & +c_{s}^{2}\delta \Upsilon \dot{\phi }=c_{s}^{3}\left(\xi _q+\xi _\Upsilon \right) + \left[(3c_{s}^{-2}-1)\ddot{\phi }+3H(1+Q)\dot{\phi }\right] \alpha + (\dot{\alpha }+ c_{s}^{2} \kappa )\dot{\phi }- \frac{3f^{\prime }}{2f^{2}}(1-c_{s}^{2})(1-c_{s}^{-2})\alpha ,\) 
